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29. MULTIWAVELENGTH PERIOD-LUMINOSITY RELATIONS

In Madore & Freedman (1991) we published fiducial PL relations in seven bandpasses: BVRIJHK. These were all based on selecting self-consistent sets of previously published LMC Cepheid data, scaled to an LMC true distance modulus of 18.50 mag and applying a single line-of-sight reddening correction using E (B-V) = 0.10 mag. Thirty-two stars were available for a calibration of BVRI PL relations; 25 stars were used for an alternative set of BVRIJHK calibrations. In the following we compare those multiwavelength PL relations with the Hipparcos sample of Galactic Cepheids, individually corrected for foreground reddening and scaled to their geometric parallax distances. We have collected from the literature multiwavelength (BVIJHK) mean magnitudes for as many of the Hipparcos-calibrating Cepheids as have been published (notably for the infrared Wisniewski & Johnson 1968, Welch et al. 1984, Laney & Stobie 1992 and reference therein). These form rather disjoint subsets. After eliminating the suspected overtone pulsators listed by FC97, the total available sample with parallaxes drops from 26 to 20. Of these only 7 have mean magnitudes published at all six wavelengths, while 10 and 13 Cepheids, respectively have either BVIJK or BVJK magnitudes in common. We have analyzed these four groups of stars independently, but self-consistently, in the following way.

Using the Hipparcos parallaxes and Galactic reddenings adopted by FC97 from Fernie, Kamper & Seager (1993) scaled to the various wavelengths using the extinction law of Cardelli et al. (1989), we derived absolute magnitudes for each of the Cepheids in each of the observed wavelengths. (We note that these corrections for interstellar extinction are not inconsiderable, ranging up to 2 mag in B for several stars). The resulting PL relations are shown in the six panels of Figure 27. Error bars are one-sigma uncertainties from the quoted parallaxes. Note the highly correlated nature of the individual data points about the fiducial lines. And too, we remind the reader that the computation of distances and their related errors from observed parallaxes is non-trivial (Brown et al. 1997), as distances are not linearly related to parallaxes, and parallax errors can subtly bias samples. A full treatment of this issue is beyond the scope of this paper, but we note that selection biases at least are minimized for stars having the smallest reported errors. As discussed by Brown et al., given the true parallax distribution the expected biases follow naturally; however, corrections to the observed parallaxes require assumptions about the true distribution, and detailed modeling. Fortunately for this application the Cepheid sample is not parallax-selected; the objects being chosen in advance based on their optical variability, periods and apparent magnitudes.

Figure 27 Figure 27. Multiwavelength Period-Luminosity relations for Cepheids with Hipparcos parallaxes, plotting all stars that have data available at the particular wavelength, as noted in the upper left corner of each panel. In each panel the solid sloping line is not a fit to the data, but rather it is the published calibration of Madore & Freedman (1991) flanked by thin parallel lines representing the 2-sigma limits quoted by them as being the intrinsic width of the instability strip at each wavelength.

The differences between these individual (trigonometric) absolute magnitudes and the predicted BVIJHK magnitudes derived from the mean PL relations of MF91 (solid lines in Figure 27) are each plotted in Figure 28 against the corresponding B-band residual. The (B-V) intrinsic color residuals are plotted against the B-band residuals in the upper right panel. The individual residuals at a given wavelength contain random contributions from the parallax uncertainties, reddening errors, and finally the intrinsic (temperature-induced) magnitude residuals which reflect the finite width of the Cepheid instability strip. The observed residuals are however extremely large (nearly 5 mag peak-to-peak) and are almost certainly dominated by the (achromatic) errors in the parallaxes, given the strict unit-slope correlation of the mag-mag residuals, and the total lack of any correlation between the magnitude-color residuals (Figure 28).

Figure 28 Figure 28. B-band residuals from the multiwavelength Period-Luminosity relations in Figure 27 are sequentially plotted as a function of residuals from each of the other five PL relations and (upper right panel) against the (B-V) color residuals. The total lack of correlation in the latter instance is unexpected except in the limit where the residuals are dominated by distance errors in the derived parallaxes. This latter situation is apparently the case given the strong (unit-slope) correlations of the residuals in each of the other panels, regardless of wavelength.

Wavelength-dependent offsets between the six mean solutions independently will reflect (1) errors in the adopted true distance to the LMC (which set all of the zero points in the MF91 multiwavelength PL relation calibrations), (2) reddening errors in the adopted extinction to the LMC sample of calibrating Cepheids, and finally (3) intrinsic differences between the LMC and Galactic Cepheids, for example, due to metallicity.

Our first solution considers the largest data set (in terms of parallaxes) but the one that is most restricted in terms of wavelength coverage: it consists of 19 Cepheids observed in B and V. Weighted by the square of the signal-to-noise ratio in the Hipparcos parallax, the residuals were summed and averaged at each of the two wavelengths giving mean offsets between the LMC calibration and the Galactic Cepheids. The variance in each mean offset was then calculated from the average of the squares of these same residuals again inversely weighted by the variance in the individually quoted parallaxes. The differences are Delta B = +0.23 ± 0.35 mag and Delta V = +0.16 ± 0.28 mag, in the sense that the LMC Cepheid calibration appears to be too faint with respect to the Galactic calibration. (Further restricting the sample to only those 12 stars with pi / sigmapi > 2.0 changes Delta B to +0.22 ± 0.24 mag and Delta V to +0.15 ± 0.17 mag.)

If the (statistically marginal, but apparently systematic) differences in the B and V solutions were to be ascribed to reddening alone, then the Galactic data and the LMC calibration can be reconciled by invoking an increase of DeltaE (B-V) = 0.07 mag in the adopted mean reddening to the LMC Cepheid sample. This is consistent with a similar suggestion regarding the LMC Cepheid calibration made recently by Bohm-Vitense (1997) based on different data. This reddening solution has the consequence that it would also require the distance modulus of the LMC to be revised downwards by -0.06 mag to 18.44 mag; the uncertainty on this offset being at least as large as the uncertainty in the individual moduli (± 0.3 mag), depending on the degree of correlation in those cumulative uncertainties. This particular path, of a reddening solution, cannot be considered definitive. Other possibilities are: (1) the LMC true modulus should be increased by (0.23 + 0.16) / 2 = +0.20 mag, without any change to the foreground reddening, or (2) that there are differential metallicity corrections amounting to -0.23 and -0.16 mag that need to be applied at the B and V wavelengths, respectively. Of course, any suitably contrived linear combination of the above three effects could also be invoked. More constraints on the problem are obviously needed.

An alternative possibility is that some of the wavelength-dependent effects seen in the comparison of Galactic (high metallicity) data with the LMC (lower metallicity) data could be due to chemical composition differences between the two samples. Taken at face value the dependence of the apparent V modulus on metallicity would be very large, Delta V / Delta[Fe / H] = 0.16 / 0.15 = 1.1 (± 1.9) mag/dex, assuming that the full offset in V noted in the above comparison is due to metallicity, and adopting a metallicity underabundance of 1.4x between the LMC and the Solar neighborhood (see, for example, FW87). However, we note that this effect is basically indistinguishable from reddening in its form (as evidenced by our first set of solutions), and that the offset (whatever its origin) when treated as reddening leads to a true distance modulus for the LMC that is unchanged, from previous assumptions, at 18.50 mag. Given this apparent degeneracy between reddening and metallicity, and the current large uncertainties in the parallaxes, assessing the dependence on metallicity from these data alone will remain problematic.

To obtain added leverage on the solution, moving to the infrared has numerous well known advantages, as first articulated in McGonegal et al. (1982): reddening effects are known to decrease with wavelength, in a well defined and calibrated manner; and simultaneously, metallicity effects are also expected to decrease in amplitude with increased wavelength.

Our second solution is based on 13 Cepheids each having BVJK data in common. This four-color solution gives a derived reddening increase for the LMC Cepheid sample of +0.04 ± 0.08 mag, with no formal offset in the derived 18.50 ± 0.13 mag true modulus for the LMC. Our next approximation employs 10 Cepheids each now having BVIJK mean magnitudes. Here the formal solution for the true modulus for the LMC is 18.53 ± 0.14 mag, with a corresponding increase in the mean reddening of +0.06 ± 0.07 mag. Finally, we have analyzed a sample of 7 Galactic Cepheids, each having BVIJHK photometry, to obtain one last solution: DeltaE (B-V) = 0.07 ± 0.07 mag with (m-M) LMC = 18.57 ± 0.11 mag. The fit to this final set of observations is shown in Figure 29; the chi2 weighted residual fitting surface being shown as an inset. The individual apparent moduli discussed here, and their errors, are summarized in Table 1.

Figure 29 Figure 29. Apparent modulus plots for LMC Cepheids observed at BVIJHK scaled to the Hipparcos zero point and using the published multiwavelength PL solutions of Madore & Freedman (1991). The solid line is a weighted chi2 fit of a reddening line to the data; the broken line indicates the one-sigma limits on that solution. Inset (top left) shows the chi2 surface indicating the minimization solution for the modulus and reddening and the interdependence of their associated errors.

 

Table 1. Multiwavelength Reddening Solutions
No.Stars µB ± sigma µV ± sigma µI ± sigma µJ ± sigma µH ± sigma µK ± sigma
19 18.73 ± 0.35 18.66 ± 0.28 . . . . . . . . . . . .
13 18.71 ± 0.36 18.64 ± 0.24 . . . 18.44 ± 0.23 . . . 18.54 ± 0.13
10 18.74 ± 0.36 18.67 ± 0.24 18.71 ± 0.20 18.44 ± 0.24 . . . 18.57 ± 0.14
7 18.86 ± 0.36 18.74 ± 0.24 18.77 ± 0.24 18.62 ± 0.18 18.60 ± 0.15 18.59 ± 0.15

Finally, if we now adopt the metallicity correction of Delta V = 0.04 mag advocated by FC97 and assume that the effects at JHK are negligible, (and eliminate B and I from the solution given that metallicity corrections for these filters are not well defined at this time) we find for this 4-color solution DeltaE (B-V) = 0.06 ± 0.11 mag with (m-M) LMC = 18.57 ± 0.11 mag. This is virtually indistinguishable from the full BVIJHK solution given above.

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